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# There are two fuses in an electrical device. Let X denote ISBN: 9780073401331 38

## Solution for problem 11SE Chapter 2

Statistics for Engineers and Scientists | 4th Edition

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Problem 11SE

There are two fuses in an electrical device. Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). Assume the joint probability density function of X and Y is a. Find P(X ≤ 2 and Y ≤ 3).

b. Find the probability that both fuses last at least 3 years.

c. Find the marginal probability density function of X.

d. Find the marginal probability density function of Y.

e. Are X and Y independent? Explain.

Step-by-Step Solution:

Step 1 of 4 :

There are two fuses in an electrical device . Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). And the joint probability density function of X and Y is We have to find

P(X ≤ 2 and Y ≤ 3).The probability that both fuses last at least 3 years.  The marginal probability density function of X.  The marginal probability density function of Y.  Check X and Y are independent.

Step 2 of 4:

Here X denote the lifetime of first fuse. And Y denote the lifetime of second fuse.

So to find  P(X ≤ 2 and Y ≤ 3) we have to integrate the  joint density function in the given     range.

P(X ≤ 2 and Y ≤ 3)  = = =    1/6 [ / -1/3 dx

=     - 1/2 [ [ ] dx

=       ( -1 =      0.3995.

(b)  The probability that both fuses last at least 3 years = P(X , Y )

P(  ,  )  =  f(x,y) dx dy

= =   -1/2  dx

=   [ = =  0.0821.

Step 3 of 4

Step 4 of 4

##### ISBN: 9780073401331

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